Transition state quantum

In zeolites, this barrier is even higher. As discussed in Section II.B, the lower acid strength and the interaction between the zeolitic oxygen atoms and the hydrocarbon fragments lead to the formation of alkoxides rather than carbenium ions. Thus, extra energy is needed to transform these esters into carbonium ionlike transition states. Quantum-chemical calculations of hydride transfer between C2-C4 adsorbed alkenes and free alkanes on clusters representing zeolitic acid sites led to activation energies of approximately 200 kJ/mol for isobutane/tert-butoxide (29), 230-305 kJ/mol for propane/sec-propoxide, and 240 kJ/mol for isobutane/tert-butoxide (32), 130-150 kJ/mol for ethane/ethene (63), 95-105 kJ/mol for propane/propene, 88-109 kJ/mol for isobutane/isobutylene, and... 

A fit of the density by a sum of terms KTpT(fs), analogous to the fit for 7 = 0, identifies 20 features up to 1.7 eV, and 15 of these are labeled in Fig. 2. Assigning transition state quantum numbers to the fitted features was simplified by analyzing each parity block separately, as described next. 

The frequency factor for propagation of a macroradical calculated as above is in good accord with the best current experimental data the transition state/quantum calculation yields 4x10 ... 

A theoretical study, using DPT, has shown that in the chlorination of toluene, ortho-, meta-, and para-sigma complexes may interconvert via a r-complex structured transition state. (Quantum chemical studies of the gas-phase chlorination of benzene, triazine. 

If reliable quantum mechanical calcnlations of reactant and transition state stnictures in vacnnm are feasible, treating electrostatic solvent effects on the basis of SRCF-PCM rising cavity shapes derived from methods... 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Voth G A, Chandler D and Miller W H 1989 Rigorous formulation of quantum transition state theory and its dynamical corrections J. Chem. Phys. 91 7749... 

Voth G A 1990 Analytic expression for the transmission coefficient in quantum mechanical transition state theory Chem. Phys. Lett. 170 289... 

Voth G A 1993 Feynman path integral formulation of quantum mechanical transition state theory J. Phys. Chem. 97 8365... 

Miller W H 1975 Semiclassical limit of quantum mechanical transition state theory for nonseparable systems J. Chem. Phys. 62 1899... 

Makarov D E and Topaler M 1995 Quantum transition-state theory below the crossover temperature Phys. Rev. E 52 178... 

Stuchebrukhov A A 1991 Green s functions in quantum transition state theory J. Chem. Phys. 95 4258... 

Shao J, Liao J-L and Poliak E 1998 Quantum transition state theory—perturbation expansion J. Chem. Phys. 108 9711 Liao J-L and Poliak E 1999 A test of quantum transition state theory for a system with two degrees of freedom J. 

Miller W H 1974 Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants J. Chem. Phys. 61 1823-34... 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

Only in the high-energy limit does classical statistical mechanics give accurate values for the sum and density of states tenns in equation (A3.12.15) [3,14]. Thus, to detennine an accurate RRKM lc(E) for the general case, quantum statistical mechanics must be used. Since it is difficult to make anliannonic corrections, both the molecule and transition state are often assumed to be a collection of hannonic oscillators for calculating the... 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

Figure A3.12.9. Comparison of the unimolecular dissociation rates for HO2—>H+02 as obtained from the quantum mechanical resonances open circles) and from variational transition state RRKM step...

Detailed analyses of the above experiments suggest that the apparent steps in k E) may not arise from quantized transition state energy levels [110.111]. Transition state models used to interpret the ketene and acetaldehyde dissociation experiments are not consistent with the results of high-level ab initio calculations [110.111]. The steps observed for NO2 dissociation may originate from the opening of electronically excited dissociation chaimels [107.108]. It is also of interest that RRKM-like steps in k E) are not found from detailed quantum dynamical calculations of unimolecular dissociation [91.101.102.112]. More studies are needed of unimolecular reactions near tln-eshold to detennine whether tiiere are actual quantized transition states and steps in k E) and, if not, what is the origin of the apparent steps in the above measurements of k E). 

Miller W H 1976 Importance of nonseparability in quantum mechanical transition-state theory Acc. Chem. Res. 9 306-12... 

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

This is a question of reaction prediction. In fact, this is a deterministic system. If we knew the rules of chemistry completely, and understood chemical reactivity fully, we should be able to answer this question and to predict the outcome of a reaction. Thus, we might use quantum mechanical calculations for exploring the structure and energetics of various transition states in order to find out which reaction pathway is followed. This requires calculations of quite a high degree of sophistication. In addition, modeling the influence of solvents on... 

HyperChem can calculate transition structures with either semi-empirical quantum mechanics methods or the ab initio quantum mechanics method. A transition state search finds the maximum energy along a reaction coordinate on a potential energy surface. It locates the first-order saddle point that is, the structure with only one imaginary frequency, having one negative eigenvalue. 

Note that the upper state quantum number of a transition is given first and the lower state quantum number second. 

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